$$f ( x ) = x - 4 - \cos ( 5 \sqrt { x } ) \quad x > 0$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [2.5, 3.5]
[0pt] - Use linear interpolation once on the interval [2.5, 3.5] to find an approximation to \(\alpha\), giving your answer to 2 decimal places.
(ii)
$$\operatorname { g } ( x ) = \frac { 1 } { 10 } x ^ { 2 } - \frac { 1 } { 2 x ^ { 2 } } + x - 11 \quad x > 0$$ - Determine \(\mathrm { g } ^ { \prime } ( x )\).
The equation \(\mathrm { g } ( x ) = 0\) has a root \(\beta\) in the interval [6,7]
- Using \(x _ { 0 } = 6\) as a first approximation to \(\beta\), apply the Newton-Raphson procedure once to \(\mathrm { g } ( x )\) to find a second approximation to \(\beta\), giving your answer to 3 decimal places.